MEDIDA DE ABERTURA DE LAS BOLSA 1. Bolsas con orificios preestablecidos

The Multinomial logit model 3.3.1

My dependent variable consists of four categorical outcomes that cannot be naturally ordered. The most appropriate regression method is therefore the multinomial logit model (MNLM). The model computes separate binary logit estimations for each pair of outcomes (Long and Freese, 2006: 223-224). Reducing the outcome to a binary model with a dichotomous

outcome would simplify the analysis, but this would waste valuable information. The MNLM uses maximum likelihood to estimate a likelihood function. It iterates until the maximum likelihood is reached (Long and Freese, 2006: 76).

Models with categorical outcomes are per definition nonlinear. It’s therefore crucial to understand the meaning of nonlinearity in order to interpret the results of my models (Long and Freese, 2006: 113). In linear models, the effect of a change in a given independent variable on the dependent variable is not conditioned on the value of that variable when it starts to change or on the values of other independent variables in the model. In nonlinear models, a change in the variable will have different effects according to its value when it starts to change. The effect is also dependent of the level of the other variables (Long and Freese, 2006: 115-116). The most common approach to nonlinear interpretation is predictions, marginal effects and creating meaningful “profiles” where predicted values can be presented. Long and Freese (2006, 118) strongly suggest applying a variety of methods in order to illustrate the results of a nonlinear model.

Assumptions and requirements 3.3.2

Sample size

In ML models, generally a minimum sample of 100 is needed, while a sample size over 500 is sufficient in most cases. More specifically, a rule of thumb suggests at least 10 observations per parameter (Long, 1997: 54). But it’s desired to have 20 observations per parameter. My

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first analysis consists of 445 observations while the second has 285. The full model has seven variables. Conducting a multinomial logit with a dependent variable with four values and seven variables amounts to 21 parameters. This should satisfy the absolute requirement, but is close to being too small.

Independence among variables

In multinomial logit, it is not a requirement that variables are statistically independent. However, large correlations may create problems for the estimates and the interpretations. To test whether there are serious problems with multicollinearity, a variance inflation factors (VIF) test can be carried out. Table 1 in Appendix B lists the VIFs of the independent

variables. A rule of thumbs suggests that VIF should be under 10 or have a tolerance (1/VIF) over .10. It also suggests that a VIF over 5 should be examined (Midtbø, 2012: 129). The variables with highest correlations are “hunger strike frequency” (6.66) and the dummy variable United States (7.56). We can anticipate that these two variables are correlated with each other. Table 2 with cross-correlations in Appendix B confirms that they are highly correlated (0.9061). I will solve this by only applying the two variables separately. Apart from this, the highest correlations in Table 2 are the correlations between Polity2 and “hunger strike frequency” (0.4224) and between “hunger strike frequency” and United States (0.3978). But as they are under 5, the rule of thumbs suggests that they can be applied in the same model. This means that multicollinearity is not a concern for my analysis.

Independence of irrelevant alternatives

An important assumption in multinomial modeling is the independence of irrelevant

alternatives (IIA). The IIA assumes that the log-odds of a given outcome is independent of the availability of other outcomes (Long and Freese, 2006: 243). McFadden (1973) said that multinomial or conditional logit only should be used if the alternatives “can plausibly be assumed to be distinct and weighted independently in the eyes of each decision-maker”.

Hunger strike outcomes are not strictly choices but results of choices made by hunger striker and authorities and government during the hunger strike. The results of these choices add up to the four possible outcomes in this thesis as illustrated in Table 2.2. The government can choose to grant a concession or to interfere with physical force to stop the hunger strike. The hunger striker can choose to surrender at any time. If none of these three choices are taken, the hunger striker will eventually die of starvation.

The IIA assumes that the log-odds of a given outcome don’t change if the number of available outcomes is reduced or expanded. Since the four hunger strike outcomes in this

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thesis are operationalized with the purpose of including all cases, it’s not easy to come up with more alternatives. However there’s a chance that in some situations the available outcomes are reduced to three. If for example a country has decided to outlaw the use of forcible feeding. We can anticipate that this would increase the chances that a hunger striker in prison will die or gain concessions, since there’s now no way for a government to thwart death except granting a concession acceptable for the hunger striker.

It’s possible to test whether the IIA assumptions holds using Hausman-McFadden (HM) test and the Small-Hsiao (SM) test by running tests with and without a reduced set of alternatives. The SM test is likely to give better results in smaller datasets and therefore the preferable among them. However, in many cases these two tests give answers that are contradictory, and their use are therefore not encouraged by Long and Freese (2006: 244).

I have nevertheless carried out the tests.7 Results from both tests are listed in

Appendix B. The Hausman test does not reject the hypothesis that the IIA assumption holds, while the Small-Hsiao test rejects the hypothesis.

Scaling, centering and transformation

Long and Freese (2006: 77) warn against problems of interpreting ML estimations if the data is not “cleaned”. If one variable contains a very high interval compared to another, this will cause large ratios between the smallest and largest standard deviations.

What’s most important for quantitative research is to separate what’s important from what’s unimportant. Therefore its crucial to not only measure direction and significance, but also the size of the effect (Ziliak and McCloskey, 2004). The variable hunger strike

frequency, ranging between 1 and 358 is therefore divided on 100 so it is more comparable to the other variables. The other variables are centered on the mean so that the intercept can be interpreted as the average probability (Stock and Watson, 2012: 152).

Tests shows that both continuous variables Length and Size fit the description of non- normality according to and are therefore transformed into their natural logarithms. This will help mitigate or eliminate both potential problems of skewness and heteroscedasticity (Wooldridge, 2009: 191).

Clustered data

Whenever a group of observations are to be considered as a subset of other observations, we are dealing with clustered (also known as hierarchical) data (Steenbergen and Jones, 2002:

7 _{The Hausman test is not compatible with clustered standard errors. I have therefore run both tests on the basis}

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219). As I have presented in Chapter 2, hunger strike outcomes are the results of both

country-specific factors and factors concerning the individual hunger strikes. As illustrated in section 2.1, the hunger strike process interlinks the protester with civil society, the public and the government. It’s been noted by Scanlan et al. (2008: 313) that “Hunger strikes (…) bridge micro- and macro-structural processes”, because they are “…relevant to many facets of social movement research including movement emergence, policing, tactical repertoires, and social movement success”. I will therefore argue that hunger strikes are natural subsets of the countries where they take place, firstly because of the political and societal surroundings of each country by which every hunger strikes must accommodate, and secondly because of the characteristics of the governments in which the hunger strikes aims to get concessions from. Since much of what we study is naturally multilevel we should apply statistical model that are also multilevel (Luke, 2004: 4). Whenever researchers aim to show causal connections

between factors operating at different levels, a multilevel analysis is the desired method (Luke, 2004: 22-23).

However, when running the intercept-only model (also called empty model) it shows a very tiny intra-class correlation coefficient (ICC) (ICC = 0.000, Std. Err = 0.013). The ICC decomposes the variance in the dependent variable and tells us the proportion of the variance that can be explained between groups (clusters). In this case, it tells how much of the variance in hunger strike outcomes that can be explained within countries. When this proportion equals to zero, it means there is no cluster variance to explain using a multilevel analysis (Hox, 2010: 56, Luke, 2004: 18). I will therefore use the multinomial logit model with clustered standard errors, as suggested by Stock and Watson (2012: 406).

Analysis strategy 3.3.3

A lesson that is often taught is that adding variables in a model should be guided strictly by theory (King et al., 1994: 182-183). Having too many variables in a statistical model will end up explaining a lot of the variance in the model but without showing any logical causal connections. This approach is often called the “kitchen sink” approach (Collier and Brady, 2010: 6). This points can be even more important in models with many parameters, such as the multinomial logit model. That said, having to explain four outcomes, also demands more explanatory power. Therefore, I have to balance this ideal against the need to have a model that manages to distinguish between my outcomes.

In order to make the most out of the limited data, I will conduct two analysis with different sample sizes. Because of the missing data on the important variables “size” and

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“length” I will first run four models (Model 1 to 4) without these two variables, testing all other variables on a larger sample (N = 445) which will give the most reliable estimates of the effects. When I then go over to Model 5 to 7, which include size and length in a smaller sample (N = 285), the four first models will be used as a reference of reliability for

interpreting any eventual changes caused by reducing the sample. A weakness in reducing the sample size is that these models cannot be meaningfully compared with the prior models through LR tests. However, the variables’ directions, sizes, as well as significance will indicate whether the smaller sample models are as valid as the large sample models.

The models will be presented step by step, adding predictors and checking for their explanatory power and significance one model at the time. This way ensures that we manage to observe how the variables act and interact with each other. This is especially important here when not only dealing with a relatively small dataset but having many parameters and dealing with a relatively unexplored research topic where both the size and directions of the effects are unknown. Having seven models also makes it possible to measure the variables’

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Analysis

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In this chapter I see how well the empirical data in the hunger strike dataset fit my theoretical models. Firstly, I will present some descriptive statistics. Then I present the estimates from the multinomial logit regression and compare the measures of fit of variables of models through different tests. Lastly, I apply various post-estimation methods in order to present the findings and what they really mean. A conclusion on the findings is presented in Chapter 5.

Multinomial logit by default sets the outcome with most observations as the base category, also called reference group. The base category is the reference to which the other outcomes are compared. The most common hunger strike outcome in the dataset is

Concession (40 %), followed by Surrender (38 %). I choose to set Surrender as the baseline outcome because the other three outcomes make more sense intuitively to explain. However, this choice only determines how the coefficient’s matrix looks and does not matter for the results.

Descriptive statistics